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Synthetic Integral Cohomology in Cubical Agda

Authors Guillaume Brunerie, Axel Ljungström, Anders Mörtberg



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Author Details

Guillaume Brunerie
  • Independent researcher, Stockholm, Sweden
Axel Ljungström
  • Department of Mathematics, Stockholm University, Sweden
Anders Mörtberg
  • Department of Mathematics, Stockholm University, Sweden

Acknowledgements

The authors are grateful to the anonymous reviewers for their insightful comments and feedback. The authors would also like to thank Evan Cavallo for many discussions and for his contributions of various useful results to the Cubical Agda library.

Cite AsGet BibTex

Guillaume Brunerie, Axel Ljungström, and Anders Mörtberg. Synthetic Integral Cohomology in Cubical Agda. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 11:1-11:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CSL.2022.11

Abstract

This paper discusses the formalization of synthetic cohomology theory in a cubical extension of Agda which natively supports univalence and higher inductive types. This enables significant simplifications of many proofs from Homotopy Type Theory and Univalent Foundations as steps that used to require long calculations now hold simply by computation. To this end, we give a new definition of the group structure for cohomology with ℤ-coefficients, optimized for efficient computations. We also invent an optimized definition of the cup product which allows us to give the first complete formalization of the axioms needed to turn the integral cohomology groups into a graded commutative ring. Using this, we characterize the cohomology groups of the spheres, torus, Klein bottle and real/complex projective planes. As all proofs are constructive we can then use Cubical Agda to distinguish between spaces by computation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constructive mathematics
  • Theory of computation → Type theory
Keywords
  • Synthetic Homotopy Theory
  • Cohomology Theory
  • Cubical Agda

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